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Fuzzy Sets and Systems 257 (2014) 85–101 www.elsevier.com/locate/fss

A discussion on fuzzy cardinality and quantification. Some applications in image processing J. Chamorro-Martínez a , D. Sánchez a,b,∗ , J.M. Soto-Hidalgo c , P.M. Martínez-Jiménez a a Dept. Computer Science and Artificial Intelligence, University of Granada, C/ Periodista Daniel Saucedo Aranda s/n, 18071 Granada, Spain b European Centre for Soft Computing, Edificio Científico-Tecnológico, 33600 Mieres, Asturias, Spain c Department of Computer Architecture, Electronics and Electronic Technology, University of Córdoba, Spain

Available online 30 May 2013

Abstract In this paper we discuss on some different representations of the cardinality of a fuzzy set and their use in fuzzy quantification. We have considered the widely employed sigma-count, fuzzy numbers, and gradual numbers. Gradual numbers assign numbers to values of a relevance scale, typically [0, 1]. Contrary to sigma-count and fuzzy numbers, they provide a precise representation of the cardinality of a fuzzy set. We illustrate our claims by calculating the cardinality of the fuzzy set of pixels that match a certain fuzzy color in an image. For that purpose we consider fuzzy color spaces previously defined by the authors, consisting of a collection of fuzzy sets providing a suitable, conceptual quantization with soft boundaries of crisp color spaces. Finally, we show the suitability of our approaches to fuzzy quantification for different applications in image processing. First, the calculation of histograms. Second, the definition of the notion of dominant fuzzy color, and the calculation of the degree to which we can say that a certain color is dominant in an image. © 2013 Elsevier B.V. All rights reserved. Keywords: Representation by levels; Fuzzy cardinalities; Color histograms; Dominant color; Gradual numbers; Fuzzy color spaces

1. Introduction The usual framework for fuzzy quantification extends quantification based on the quantifiers ∃ and ∀, by allowing linguistic fuzzy quantifiers. In Zadeh’s approach [55] two kinds of quantifiers are considered: absolute quantifiers, corresponding to fuzzy subsets of the non-negative integers of the form around n or approximately between n and n , and relative quantifiers, corresponding to fuzzy subsets of the real interval [0, 1], representing imprecise percentages like around a fraction q, approximately more than a fraction q, etc. These quantifiers can be seen as fuzzy numbers defined as normalized, convex fuzzy sets defining restrictions on their respective domains. Other quantifiers have been also proposed following the theory of generalized quantifiers (TGQ) [3,27] which recognizes more than 30 types of quantifiers, but will not be discussed here. * Corresponding author at: Dept. Computer Science and Artificial Intelligence, University of Granada, C/ Periodista Daniel Saucedo Aranda s/n, 18071 Granada, Spain. E-mail addresses: [email protected] (J. Chamorro-Martínez), [email protected] (D. Sánchez), [email protected] (J.M. Soto-Hidalgo), [email protected] (P.M. Martínez-Jiménez).

0165-0114/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.fss.2013.05.009

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Fuzzy quantifiers are very useful in the setting of Computing with Words in order to generate linguistic statements about the number or percentage of objects that verify certain properties. Absolute quantifiers are employed in the so-called Type I sentences, with the form “Q of X are A”, where Q is an absolute quantifier, X is a crisp set, and A is a fuzzy subset of X. Relative quantifiers are employed in Type II sentences, with the form “Q of D are A”, where Q is a relative quantifier and D is a fuzzy subset of X. Linguistically quantified sentences have been applied in a large amount of applications like data mining and data fusion, fuzzy control, fuzzy expert systems, decision-making, fuzzy queries in databases, linguistic summarization, fuzzy description logics, etc. We are interested in their application in the area of image processing, as a tool for helping to fill the semantic gap between the storage of images in computers and their description in terms of perceptual concepts employed by human beings. Many authors have employed fuzzy sets for representing semantic concepts like colors, texture features, and shapes. The notion of region is also perceptual, corresponding to a subset of pixels that is connected and homogeneous with respect to some semantic concept (or some combination of them), and several authors have argued that its most suitable representation is by means of fuzzy subsets of pixels, the so-called fuzzy regions. On this basis, other semantic concepts can be defined by means of quantified sentences. Some examples are: • A fuzzy region is large when At least a fraction q of the pixels in the image are in the region, for some appropriate, user-defined value q ∈ [0, 1]. This is a type II sentence because the quantifier At least a fraction q is relative. The set D is the set of pixels in the image (a crisp set in this particular case), and the set A is the fuzzy region. • A fuzzy region is mostly red when At least a fraction q  of the pixels in the region are red, for some appropriate q  ∈ [0, 1]. Here, the quantifier is At least a fraction q  , the set D is the fuzzy region, and the set A is the fuzzy set of pixels in the image that are red, where red is a fuzzy color, i.e., a fuzzy subset of crisp colors corresponding to our perception of red [41]. The set A can be easily obtained by computing the membership of each pixel’s crisp color to the fuzzy set red. • A fuzzy color is dominant in an image when At least a fraction q  of the pixels in the image are red, for some appropriate q  ∈ [0, 1]. • A fuzzy region is mostly in the center of an image when most of the pixels of the region are in the center of the image. Here, the quantifier is most, the set D is the fuzzy region, and the set A is a fuzzy set of pixels with maximum membership in the geometrical center of the image and decreasing to the borders, that can be defined in different ways. The fulfilment of quantified sentences is a matter of degree, and hence the semantic concepts that we can define using them are also fuzzy concepts. This is natural since quantifiers and the sets D and A are fuzzy sets. The accomplishment degree of quantified sentences can be calculated as the compatibility between the restriction defined by the quantifier, and a measure of the absolute cardinality of A (resp. relative cardinality of A with respect to D) for type I (resp. type II) sentences [55,16]. Hence, in order to develop methods for obtaining a reliable accomplishment degree, it is necessary to obtain good measurements of the absolute and relative cardinality of fuzzy sets. Several authors have proposed ways to measure the cardinality of a fuzzy set, extending the classic one in different ways (see different compilations in [43,15,47]). Some of them have been employed for calculating the accomplishment of quantified sentences [55,51,16]. The most common approaches are the scalar cardinality and the fuzzy cardinality of a fuzzy set. The first approach claims that the cardinality of a fuzzy set is measured by means of a scalar value, either integer or real, whereas the second approach assumes the cardinality of a fuzzy set is just another fuzzy set over the non-negative integers. Among the latter, it is common to consider that the cardinality of a fuzzy set must be a fuzzy number, i.e., normalized and convex. However, this point has also been criticized [15]. Recently, an alternative called gradual numbers has been introduced by Dubois and Prade [21], that has been also employed in the evaluation of quantified sentences [31,37,38]. In this paper we discuss on some of the aforementioned representations of the cardinality of a fuzzy set and their use in fuzzy quantification. We show that fuzzy numbers are the best choice for representing restrictions like linguistic quantifiers, and their arithmetics is that of restrictions. On the other hand, gradual numbers are the best choice as measures of the cardinality of fuzzy sets. Hence, the evaluation of the accomplishment of quantified sentences is to be performed by calculating the compatibility between a fuzzy number and a gradual number. We illustrate our claims by calculating the cardinality of the fuzzy set of pixels that match a certain fuzzy color in an image. For that purpose we consider fuzzy color spaces previously defined by the authors, consisting of a collection

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of fuzzy sets providing a suitable, conceptual quantization with soft boundaries of crisp color spaces [41]. Finally, we show the suitability of our approaches to fuzzy quantification for different applications in image processing. First, the calculation of histograms, in which we have done some previous work in [10,11]. Second, the definition of the notion of dominant fuzzy color, extending our previous work in [12], and the calculation of the degree to which we can say that a certain color is dominant in an image. 2. Cardinality of fuzzy sets In this section we briefly describe the most widely employed approaches for measuring the cardinality of fuzzy sets, and we discuss the most employed techniques in each approach. Though we do not perform an exhaustive study of the (huge) amount of measures in the literature, our conclusions for each approach can be extended in general to all the measures within them. 2.1. Scalar cardinalities In these approaches, the cardinality of a fuzzy set is a crisp number, either real or integer [47]. A large amount of scalar measures are available in the literature, including intervals where the scalar cardinality is to be found [19,33, 44–46,7]. However, probably the most employed cardinality for fuzzy sets is the scalar sigma-count, defined for any fuzzy set F : X → [0, 1] as  sc(F ) = F (x) (1) x∈X

This can be extended to the case of relative cardinality as follows:  (F (x) ∩ G(x)) sc(F ∩ G)  sc(F /G) = x∈X = sc(G) x∈X G(x)

(2)

where the intersection is performed via the minimum. This measure is probably the most employed in the literature, and has several interesting properties: it can be computed and stored very efficiently, it satisfies sc(F ) + sc(F ) = |X|, and the arithmetics is that of real numbers. It is also easily understandable since we are used to see cardinality as a scalar number in the case of crisp sets. Going further on this, it is usual to round the sigma-count (a real number in general) to the nearest integer. However, this measure has several important drawbacks. In fact, sc is a measure of energy of a fuzzy set [32], i.e., a measure of the global membership in the set. Whilst this is somehow correlated with cardinality (larger membership is expected to be correlated to larger cardinality), this is not exactly the same. The accumulation of small degrees can yield the same cardinality as having a single element with total membership. This is the case if we consider two  to a degree 0.1, and having 10 pixels compatible situations like having 100 pixels compatible with a fuzzy color C  to a degree 1; in both cases, the result of the sigma-count applied to the fuzzy set of pixels compatible with C  with C is 10, which is not very intuitive for the first case. Some alternatives consider in addition only values above a certain threshold, but this does not solve other problems, as we shall see. The real problem with scalar measures is that they are not really suitable for providing precise information about the cardinality of a fuzzy set. Using a scalar measure of the cardinality is like using a crisp set for representing a fuzzy set, that is, we are losing information for the sake of obtaining a simpler and more easily manageable measure. The problem is that this kind of summary is not always representative, or it is losing too much information. We will come back to this later. 2.2. Fuzzy cardinalities The next approach in popularity is to consider that the cardinality of a fuzzy set is a fuzzy subset of the non-negative integers. In this category, which is considered as a better representation of cardinality by many authors as it gives more accurate information, are the definitions by Zadeh [54,55], Dubois and Prade [19], Wygralak [14,18], and Delgado et al. [15], among many others. In [15] a family of measures providing fuzzy subsets over the non-negative integers is introduced having several of the existing methods as particular cases, and introducing new ones.

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As a particular case, many authors have suggested that the cardinality of a fuzzy set must be a fuzzy number, i.e., a normalized, convex subset of the real line or the non-negative integers [19,47]. However, in [15] we showed that in some cases this is counterintuitive. Consider for example the fuzzy set given by A = 1/x1 + 0.5/x2 + 0.5/x3 . The cardinality of A could be one (because x1 belongs to A for sure) or, if we relax our criterion to accept elements in A, the cardinality could be three (accepting x2 and x3 belong to A as well). However, the cardinality cannot be two, since if x2 ∈ A then x3 ∈ A and vice versa. This way, the cardinality is not convex. In addition, this example illustrates that the sigma-count is not always a representative measure since in this case sc(A) = 2, an impossible value. Several authors have related this issue to the idea that the possible cardinalities of a fuzzy set are the cardinalities of its α-cuts, since these are the possible crisp representatives of the fuzzy set, and hence that should be the support of the fuzzy cardinality [15]. In our previous example the possible cardinalities of A are 1 or 3 since its possible α-cuts are {x1 } and {x1 , x2 , x3 }. Several alternative proposals that comply with this idea are available [54,15]. It is immediate that fuzzy cardinalities are more complex to calculate, store, and manipulate than scalar ones. In addition, arithmetics is more complex and lacks some of the properties of scalar arithmetics. In particular, when operating with fuzzy numbers, sc(F ) + sc(F ) = |X| does not hold in general, since operations use to increase the imprecision of results. The situation with fuzzy numbers is even worse for the case of relative cardinalities, as this implies to perform a quotient of absolute cardinalities. When the latter are represented by fuzzy numbers, it is difficult to perform operations leading to a convex rational result without introducing in the support values which are clearly impossible for the relative cardinality of two sets. For instance, let |X| = 5 and F be a fuzzy subset of X whose cardinality is measured as a fuzzy number 1/1 + 0.5/2 + 0.3/3. Then, the relative cardinality of F with respect to X is 1/0.2 + 0.5/0.4 + 0.3/0.6, which is not convex in the set of rational numbers since 0.3 is not in the support. This problem is solved if convexity is not imposed. But even if we assume non-convexity, there are problems regarding how to combine the different cardinalities in the support of the fuzzy sets F ∩ G and G in order to obtain the relative cardinality of F with respect to G. In the next section we show an alternative approach that solves all these problems. 2.3. Gradual numbers In [21], Dubois and Prade introduced the ideas of gradual element and gradual number as a way to represent fuzzy quantities. Gradual numbers assign numbers to values of a relevance scale, typically [0, 1]. The cardinality of a fuzzy set can be represented by a gradual number in which the cardinality of the α-cut of the fuzzy set is assigned to α. Following the notation in [36], a gradual (real) number is a pair (Λ, R) where Λ ⊂ (0, 1] is finite, and R : (0, 1] → R. Then, for a fuzzy set F , its cardinality is represented by the gradual number gc(F ) = (Λ|F | , R|F | ) where Λ|F | = {F (x) | x ∈ support(F )} ∪ {1} and for each α ∈ Λ|F | R|F | (α) = |Fα |

(3)

with Fα the α-cut of F . In a similar way, the relative cardinality gc(F /G) can be easily calculated by considering Λ|F /G| = ΛF ∪ ΛG and R|F /G| (α) =

|(Fα ∩ Gα )| |Gα |

(4)

Gradual numbers offer several advantages. First, they don’t introduce imprecision in the cardinality, since each α-cut is assigned a crisp number. On the contrary, the α-cut of a fuzzy number is an interval where the cardinality is assumed to be, and hence it is an imprecise representation of the cardinality. Hence, the so-called fuzzy numbers are in fact fuzzy intervals [20]. Another advantage is arithmetics. Let RRL be the set of gradual real numbers. Operations are extended as follows: Definition 2.1. Let f : Rn → R and let R1 , . . . , Rn be gradual numbers. Then f (R1 , . . . , Rn ) is a gradual number with  Λf (R1 ,...,Rn ) = Λ Ri (5) 1in

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and, ∀α ∈ Λf (R1 ,...,Rn )

  Rf (R1 ,...,Rn ) (α) = f RR1 (α), . . . , RRn (α)

(6)

With these operations, gradual numbers have the same algebraic structure as ordinary numbers, whilst fuzzy numbers satisfy the properties of interval arithmetics only. In addition, gradual numbers do not increase imprecision of the representation with operations, even to the extent that operations on gradual numbers may yield crisp numbers. In particular, it is always the case that for any gradual number a, a − a = 0, a/a = 1 (provided a = 0), etc. They are hence a precise and very useful representation when the cardinality is to be employed in any calculation. In order to show further properties, we have to introduce the notion of representation by levels. 2.3.1. Representation by levels Gradual numbers are a particular case of representation by levels (see [40]), developed in order to represent fuzzy mathematical objects, in particular concepts, and akin to the notion of gradual element and gradual set introduced by Dubois and Prade in [21]. In the case of concepts represented as subsets of a reference set X, a representation by levels of a fuzzy concept A is a pair (ΛA , ρA ) where again ΛA ⊂ (0, 1] is finite, and ρA : ΛA → P(X), with P(X) being the power set of X. Representation by levels and the corresponding operations are an alternative to fuzzy set for representing fuzzy concepts, with different operations and properties. We can remark: • Fuzzy sets with a finite number of different degrees of membership are particular cases of representation by levels if we consider their representation by means of a collection of α-cuts. However, contrary to fuzzy sets, in which the α-cuts are nested in the well-known way, the representation by levels does not impose any relation between the representation in every level. Hence, not every representation by levels of fuzzy concepts is a fuzzy set. • The philosophy of the representation by levels for representing fuzzy objects is to assign crisp versions of the object to a collection of levels, and to extend operations between objects by performing the operations in every level independently, as we have seen with gradual numbers (a representation by levels of numbers under fuzziness). As a particular case, the conjunction and disjunction of two fuzzy sets, considered as representation by levels, yields the usual result for the standard union and intersection via maximum and minimum. However, the negation does not yield the usual complement of fuzzy sets. In fact, the negation of a fuzzy set is NOT a fuzzy set under the representation by levels, since what is obtained is the complement of each α-cut in each level, which does not form a fuzzy set. When considering operations by levels, all properties of cardinalities of crisp sets are kept, including the valuation property gc(F ) + gc(G) = gc(F ∪ G) − gc(F ∩ G)

(7)

and the particular case gc(F ) + gc(F ) = |X|. A disadvantage of gradual numbers is related to the space needed for representing a gradual number, and the time needed for arithmetic calculations, both being proportional to the number of α-cuts employed. In practice, only a finite number of cuts are necessary, corresponding to levels in the representation of the gradual number, see [36,40]. 2.4. Discussion From the discussion above, we conclude that scalar cardinalities are not accurate representations of the cardinality in general. They can be seen as summaries of the real cardinality, either by discarding the cardinality of all but one of the α-cuts, or by providing the center of gravity, like in the case of the sigma-count (this interpretation is given in [15]). However, the summary they provide may be not representative of the cardinality as well. An important question for us is, why several authors have forced fuzzy cardinalities to be convex fuzzy sets, i.e., fuzzy numbers? In our opinion, there has been a confusion between measuring cardinality and expressing cardinality to a human user. In the crisp case, non-negative integers can be employed for both things. However, in the fuzzy case, measuring implies assuming non-convexity, as we stated above, and even discarding fuzzy numbers (since they

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introduce imprecision), whilst expressing implies discarding gradual numbers and using fuzzy numbers that are, in fact, fuzzy intervals [20] representing in the best possible way linguistic terms of the form “approximately x” or “approximately between x and y”. Hence, we do not claim that gradual numbers are better than fuzzy numbers but simply that they serve to different, complementary purposes. Gradual numbers are the most accurate measure of cardinality, and the best choice for representing cardinalities to be used in further calculations. However, gradual numbers are not as intuitive as fuzzy numbers when the cardinality is to be expressed to a human user. In particular, it is difficult to provide a linguistic label describing precisely a gradual number. On the contrary, fuzzy numbers represent restrictions, linguistic concepts on numbers, but are not well suited for providing accurate representations, since the latter are non-convex in general [15] and they introduce imprecision in the measurement. Their arithmetics is an arithmetics of restrictions (fuzzy intervals), that is not appropriate for representing cardinalities that are to be employed in further calculations. On the contrary, they are the best choice for summarizing the cardinality into a linguistic term to be provided to human users (losing information and accuracy, but less than scalar cardinalities). Hence, our proposal is to use gradual numbers for measuring cardinality and to compute with it for any purpose and, if it is necessary to give a linguistic information to a human user about the cardinality of a fuzzy set, try to find the most informative linguistic label in the vocabulary of the user (assuming every label is represented by a given fuzzy number) from those which are maximally compatible with the gradual number. It is immediate that this is equivalent to evaluating the accomplishment degree of a quantified sentence, in which the fuzzy numbers are quantifiers, and the cardinality of the fuzzy set A (or the relative cardinality of A with respect to D) is measured via gradual numbers. As we shall see in the next section, some approaches to fuzzy quantification can be seen in practice as performing this procedure. Notice that, this way, fuzzy numbers expressing cardinalities to the user are not calculated directly from the fuzzy sets, but are chosen according to their compatibility with the gradual number obtained from the fuzzy sets, which is an accurate measurement of the cardinality. 3. Fuzzy quantification In the literature there are many different methods for assessing the accomplishment degree of quantified sentences, based on different approaches (fuzzy integrals, OWA operators, etc.), see among others [55,48,19,49,5,6,33,13,22,16, 4,23,17,1,2,24,52,14,31,25,37,38]. As we indicated in the introduction, one of the possible approaches is based on measuring the cardinality of fuzzy sets. In this approach, the accomplishment degree of type I quantified sentences of the form “Q of X are A” can be calculated as the compatibility between the restriction defined by the absolute quantifier Q and a measure of the absolute cardinality of A. Similarly, the accomplishment degree of type II quantified sentences of the form “Q of D are A” can be calculated as the compatibility between the restriction defined by the relative quantifier Q and a measure of the relative cardinality of A with respect to D. The cardinality approach has been employed for instance in [55,16,37,38], and the reliability of the results depends heavily on the quality and representativeness of the measurement of cardinality. Several authors have proposed theoretical properties that should be intuitively satisfied by any evaluation method, and most of the existing methods have been assessed on the basis of these properties, see for instance [16,4,1]. In this section we show some evaluation techniques based on the three cardinality measures discussed in the previous section. We also provide a new evaluation technique inspired in the philosophy of representation by levels. 3.1. Evaluation based on the sigma-count In this kind of evaluation, proposed by Zadeh [55], cardinality is measured by the sigma-count, and the compatibility between the quantifier and the cardinality is simply the membership of the latter to the former. More specifically, the evaluation of the type I sentence “Q of X are A” is Q(sc(A)), whilst the evaluation of the type II sentence “Q of D are A” is Q(sc(A/D)). The properties of this method have been discussed in [16]. It is a very strict method for the evaluation of crisp quantifiers, in particular the evaluation for the quantifiers ∀ and ∃. Specifically, the evaluation of “∀ of X are A” is 1 iff A = X, and 0 otherwise, whilst the evaluation of “∃ of X are A” is 0 iff A = ∅, and 1 otherwise. In addition, arbitrarily small changes in the membership can change the evaluation from 0 to 1 and vice versa. Finally, as we have seen, the sigma-count is not a good measure of cardinality in general. As an example of what may happen by using

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sigma-count, suppose a fuzzy set A has in its support 100 elements with degree 0.5. Then, the possible cardinalities of the set, given by the cardinalities of its α-cuts, are 0 and 100, no other possibility. However, the sigma-count is 50, so the evaluation of the sentence “Around 50 of X are A” will be 1, when one should expect the result to be 0 under a reasonable interpretation of “Around 50”. 3.2. Evaluation based on fuzzy cardinalities In [16] we showed that some of the most employed evaluation methods for type I and type II sentences with D = X can be interpreted as particular cases of cardinality-based methods, in which the compatibility between quantifier and cardinality is calculated via different combinations of t-norms and t-conorms using the expression    Card(A)(i) ⊗ Q(i) (8) i∈support(Card(A))

with Q an absolute quantifier, ⊕ and ⊗ a t-conorm and a t-norm respectively, and Card(A) being the fuzzy cardinality of A. The expression for the case of type II sentences with D = X is similar but replacing Q(i) by Q(i/|X|) since Q is relative. Among others we showed that: • When using non-decreasing quantifiers, Yager’s evaluation method based on the use of OWA operators, and the method based on the Choquet fuzzy integral, are equivalent to the cardinality method based on the cardinality ED introduced in [15], using Lukasiewicz’s t-conorm and the product as t-norm in Eq. (8). We extended this method to any kind of quantifiers and to type II sentences with arbitrary D in [16] by defining a fuzzy measure of the relative cardinality that extends ED. The resulting method is called GD, and it is defined as follows for type II sentences:

 |(A ∩ D)αi | GDQ (A/D) = (αi − αi+1 )Q (9) |Dαi | αi ∈(A/D)

where (A ∩ D)(x) = Min(A(x), D(x)), (A/D) = Λ(A ∩ D) ∪ Λ(D), Λ(D) being the level set of D, and (A/D) = {α1 , . . . , αp } with αi > αi+1 for every i ∈ {1, . . . , p}, α1 = 1 and αp+1 = 0. The set D is assumed to be normalized. If not, D is normalized and the same normalization factor is applied to A ∩ D. • When using non-decreasing quantifiers, the method based on the Sugeno fuzzy integral is equivalent to the cardinality method based on the cardinality introduced by Dubois and Prade in [19], or Zadeh’s first fuzzy cardinality measure [54], using in both cases the maximum as t-conorm and the minimum as t-norm in Eq. (8). We extended this method to any kind of quantifiers and to type II sentences with arbitrary D in [16] by defining a suitable measure of the relative cardinality. The resulting method is called ZS, and it is defined as follows for type II sentences: 

|(A ∩ D)α | ZSQ (A/D) = max min α, Q (10) |Dα | α∈(A/D) The different properties of these approaches based on the cardinality have been studied in [16]. 3.3. Evaluation based on gradual numbers The basic idea of the evaluation based on gradual numbers is to evaluate the sentence in every level of the representation, in which the cardinality is crisp, and then to aggregate the different evaluations into a single value. Different proposals for the aggregation were provided in [31,37,38]. One specific aggregation procedure introduced both in [31] and [37] yields the method GD of Eq. (9). Following the ideas behind the representation by levels, the final aggregation is not strictly necessary, and it is performed simply to be able to provide a numerical accomplishment degree, as in classical methods. Instead, when the result of the evaluation is to be employed for further calculation, we propose to perform calculations in each level independently. Formally, the evaluation by levels is as follows [37]: the evaluation of E ≡ “Q of D are A” is a gradual number E in [0, 1], defined by (ΛE , RE ), where ΛE = ΛA ∪ ΛD , ΛE = {α1 , . . . , αm } with 1 = α1 > α2 > · · · > αm > αm+1 = 0, and, ∀α ∈ ΛE ,

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Table 1 Representation by levels of D, A, and the gradual numbers gc(A/D), E, and B for the quantifier of Eq. (15). α

ρD (α)

ρA (α)

gc(A/D)

E

B

1 0.9 0.8 0.6 0.5 0.4

{x1 } {x1 } {x1 , x2 } {x1 , x2 , x3 } {x1 , x2 , x3 } {x1 , x2 , x3 , x5 }

∅ {x1 } {x1 } {x1 , x3 } {x1 , x3 , x4 } {x1 , x3 , x4 }

0 1 1/2 2/3 2/3 1/2

0 1 0 2/3 2/3 0

0 1 0 1 1 0





  |ρA∧D (α)| |ρA (α) ∩ ρD (α)| RE (α) = Q =Q = Q R|A/D| (α) |ρD (α)| |ρD (α)|

(11)

This evaluation can be aggregated into a final value S(E), equivalent to GDQ (A/D) as explained before, as follows:  S(E) = (αi − αi+1 ) · RE (αi ) (12) αi ∈ΛE

An important difference between the proposals in [31] and [37,38] is that in the last two we consider the representation by levels as the representation of fuzzy concepts that are employed in the quantified sentences, that is, the usual framework of type I and type II sentences is extended, allowing us to consider general representations by levels of D and A, without restricting ourselves to fuzzy sets. When this is the case, the procedure obtained is no more yielding the same result as GD. Even if fuzzy sets are considered, represented by a collection of α-cuts nested in the usual way, operation of complement considered in [37,38] is that of the representation by levels paradigm, hence leading to representations which are not fuzzy sets. As explained in [37,38], this approach allows us to satisfy all the intuitive properties proposed by several authors that, in the setting of fuzzy set theory, cannot be satisfied simultaneously, see [39] for more details. However, the aforementioned approaches are not totally in accordance with the philosophy of representation by levels. To fully comply with that, quantifiers should be also represented by levels, so that in each level the quantifier will be represented by a crisp quantifier, and the evaluation in each level will be the compatibility between the crisp number and the crisp quantifier in that level, yielding a crisp evaluation (either 0 or 1). This new evaluation method we introduce here can be formalized as follows: the evaluation is a gradual number B = (ΛB , RB ) where ΛB = ΛE and, ∀α ∈ ΛB ,

1 Q(R|A/D| (α))  α (13) RB (α) = 0 otherwise Again, if we want to obtain a numerical result, we can aggregate the result using the following expression:   S(B) = (αi − αi+1 ) · RB (αi ) = (αi − αi+1 ) (14) αi ∈ΛB

{αi |Q(R|A/D| (αi ))αi }

In order to see the difference between both methods we recall the following example from [37]: consider A and D defined by the following fuzzy sets: D = 1/x1 + 0.8/x2 + 0.6/x3 + 0.4/x5 A = 0.9/x1 + 0.6/x3 + 0.5/x4 Then we have ΛD = {1, 0.8, 0.6, 0.4} and ΛA = {1, 0.9, 0.6, 0.5}. Table 1 shows the representation by levels of both sets in the union of their levels, as well as the gradual number representing the relative cardinality gc(A/D), and the gradual evaluation using the two aforementioned methods for the sentence “Qmost of D are A” with Qmost defined in Eq. (15).  0 x  0.5 Qmost (x) = 1 (15) x  0.75 4(x − 0.5) otherwise

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We can see that B provides a crisp evaluation in each level. Finally, using the aggregations previously defined, we have S(E) = 7/15 ≈ 0.46 and S(B) = 0.3. We shall study further properties of the new method in a forthcoming paper. 4. Some applications in image processing In this section we show two applications of fuzzy cardinality and quantification in image processing. As we have concluded in previous sections, the basic idea is to employ gradual numbers for measuring cardinality when the latter is to be employed in further calculations. On the contrary, when the idea is to provide information to a human user about the cardinality, the idea is to employ fuzzy quantification in order to calculate the compatibility between the cardinality, measured as a gradual number, and a collection of fuzzy numbers corresponding to linguistic quantifiers, in order to provide the most compatible one to the user. This collection of quantifiers will form the vocabulary that we consider understandable to the user. The two applications we show here are related to the cardinality of the fuzzy colors in an image, understood as the cardinality of the fuzzy subset of pixels having that fuzzy color. Hence, we shall start recalling the notion of fuzzy color space we introduced in [41]. Then we shall describe the application of fuzzy cardinality and quantification in the calculation of histograms of colors and dominant colors. 4.1. Fuzzy color spaces In order to represent the semantic compatibility between crisp colors and linguistic color terms, in [41] we introduced the following definitions of fuzzy color and fuzzy color space on a generic crisp color space XY Z with domain of components being DX , DY , and DZ :  is a linguistic label whose semantics is represented in a color space XY Z Definition 4.1. (See [41].) A fuzzy color C by a normalized fuzzy subset of DX × DY × DZ . 1 , . . . , C m that define a fuzzy partition  Definition 4.2. (See [41].) A fuzzy color space XY Z is a set of fuzzy colors C of DX × DY × DZ in the sense that it satisfies:    1. m i=1 sup(Ci ) = XY Z, i.e., the union of the supports of the Ci covers the whole space. i ) ∩ ker(C j ) = ∅ ∀i = j , i.e., the kernels of the C i and C j are pairwise disjoint, where ker(C)  = {c ∈ XY Z | 2. ker(C  = 1}. C(c) i (c) = 1, i.e., there is at least one object fully representative of the fuzzy 3. ∀i ∈ {1, . . . , m} ∃c ∈ XY Z such that C i . color C Condition 3 is always verified by definition of fuzzy color. Condition 1 implies ∀c ∈ XY Z ∃i ∈ {1, . . . , m} such    that m Ci (c) > 0. Conditions 2 and 3 imply Ci  Cj ∀i = j . Notice the usual definition of fuzzy partition given by  (c) = 1. C i i=1 In [41] we proposed several fuzzy color spaces using color names provided by the well-known ISCC–NBS system [28]. ISCC–NBS provides several color sets in the form of sets of pairs (linguistic term, crisp color). Using the methodology introduced in [41], we calculate for each color set a fuzzy color space on the basis of a Voronoi diagram of the crisp color space, calculated using the crisp colors of the set of pairs considered. The Voronoi diagram is a crisp partition corresponding to the 0.5-cut of the fuzzy colors. The kernel and support of each fuzzy color are obtained as a scaling with parameters α and β respectively, with α < 1 < β, and guaranteeing the conditions in Definition 4.2. The membership functions of the fuzzy colors are obtained on the basis of distances in the crisp color space. For more details see [41]. In [41], we have obtained three fuzzy color spaces on the basis of the sets of color names Basic (13 colors), Extended (31 colors) and Complete (267 colors) in the RGB color space. For instance, the Basic set has color names corresponding to ten basic color terms (pink, red, orange, yellow, brown, olive, green, blue, violet, purple), and 3 achromatic ones (white, gray, and black). The corresponding representative crisp colors are shown in Fig. 1, together with a rough view of the core, the α-cuts of level 0.5, and the support of some fuzzy colors in the fuzzy color space

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Fig. 1. Part of the RGB fuzzy color space obtained in [41] from the ISCC–NBS Basic set of colors. (a) ISCC–NBS Basic set of colors (representative crisp color and color name). (b) Situation of the representative crisp colors in the RGB color space. (c) Volumes of colors in the 0.5-cut for the fuzzy colors yellow, blue, green, and gray obtained from the Voronoi diagram in the RGB cube. (d) Volumes of colors in the kernel of the same fuzzy colors. (e) Volume of colors in the support of the fuzzy color yellow. (f) Superimposed views of part of the surfaces of the volumes of colors in the kernel (most internal), 0.5-cut (middle) and support (most external) for the fuzzy color yellow. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)

obtained from ISCC–NBS Basic in [41]. We are not showing examples of the fuzzy color spaces for the sets Extended and Complete because of the lack of space. 4.2. Histograms on fuzzy color spaces Histograms are the basis of many techniques for image restoration, enhancement, segmentation, retrieval, etc. In principle, a color histogram is defined as a function h(ck ) = nk where ck = [x, y, z] is a color and nk is the number of pixels in the image having the color ck . It is common to normalize a histogram by dividing each of its values by the total number of pixels, obtaining the frequency of occurrence of a color ck . This simple approach has the drawback that a crisp color space is not representative of the collection of colors we can distinguish and identify. In addition, the values nk are to be very low because there are many colors in a crisp color space, and it is easy to find small color variations in real images. Since in practice many of the colors ck are indistinguishable for us, a solution is to use an histogram defined on groups of indistinguishable colors Ck , in which the associated number of pixels is h(Ck ) = ck ∈CK nk . The collection of groups of colors C1 , . . . , Cn defines a partition (quantization) of the color space employed. For this purpose, we shall employ the fuzzy color spaces introduced in the previous section. This approach has the advantage that it is able to represent the fact that indistinguishability is a fuzzy, gradual concept for us, i.e., colors are indistinguishable to a certain degree. Crisp boundaries inherent to crisp quantization are counterintuitive for us. In addition, both the (fuzzy) quantization and the histograms are less sensitive to small variations of the boundaries. In the literature there are several proposals which define histograms over a set of fuzzy colors [18,26,34,35]. One drawback of most of these proposals is that they work only with intensities. In [10] we employed non-convex fuzzy subsets of the non-negative integers for defining fuzzy histograms. This approach represents a compromise between the advantages and disadvantages of using fuzzy numbers and gradual

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numbers. In [11] we proposed two kinds of color histograms: one based on gradual numbers, to be used in practice by the computer for image processing, and an approximation of this histogram by means of linguistic labels (fuzzy numbers) in order to give information to the user. These are introduced and illustrated in the next sections. 4.2.1. Gradual histogram Definition 4.3. A gradual color histogram is a function hG that assigns a gradual number to every fuzzy color in a  fuzzy color space XY Z.  is obtained by assigning to α the cardinality of the α-cut The gradual number corresponding to every fuzzy color C  Since the fuzzy subset of pixels with color C  is finite, in practice we need of the fuzzy subset of pixels with color C. only a finite number of cuts for representing the gradual number. However, this number may be large, so in some cases it may be interesting to consider just a fixed collection of cuts with equidistant values of α. See [36] for further discussion on that. Let us remark that the gradual integer number can be easily transformed into a gradual rational number by dividing the number associated to each level α by the total number of pixels in the image. This is convenient when we want the histogram to represent proportions instead of absolute values. We shall see examples in Section 4.2.3. Finally, usual image processing operations performed on crisp histograms can be extended directly to gradual histograms by using Definition 2.1, i.e., by applying the operation in each level. 4.2.2. Linguistic histogram  Z be a fuzzy color space. Let us consider a collection of fuzzy linguistic quantifiers SQ = {Q1 , . . . , Qk } Let XY provided by the user, like Around 10% or Most, represented by appropriate fuzzy subsets of the unit interval, and Between 50 and 100, represented by appropriate fuzzy subsets of the non-negative integers. Definition 4.4. A linguistic color histogram is a function hL that assigns a linguistic quantifier from SQ to every fuzzy  color in XY Z. We consider user-defined quantifiers in order to improve understandability, since accuracy of the linguistic approach is always worse than that of the gradual approach.  we take the gradual number associated to C  in the gradual In order to obtain the quantifier for every fuzzy color C, color histogram (an accurate representation of cardinality) and we calculate the compatibility between this gradual number and every quantifier in SQ . The quantifier that yields maximum compatibility is then chosen to represent the linguistic amount or proportion of pixels having color in the linguistic histogram. The compatibility is calculated by  using evaluating the accomplishment degree of the quantified statement Qi of pixels in the image are painted in C, one suitable evaluation method. In this paper we shall consider the method we have introduced in Section 3.3. 4.2.3. A first synthetic example Our first example, taken from [36], allows us to illustrate the differences between the different approaches to cardinality of fuzzy sets discussed in Section 2. The fuzzy color space employed here, described in [36], is not that of Section 4.1, but this is unimportant for our purpose in this first example. Fig. 2 shows two images, the first one 1 , . . . , C 8 denote these colors, from left to containing eight crisp colors in the kernel of different fuzzy colors. Let C right and top to bottom (see [36] for a definition of the membership functions). In the second one we have four crisp colors that are compatible to a degree 0.5 with two of the eight fuzzy colors in the first image. Table 2 shows the (relative) cardinality of the fuzzy set of pixels painted in each fuzzy color in both images, using different approaches. The result is the same for all the fuzzy colors in both images by the way they have been defined. In the case of image A, since the eight crisp colors employed are in the kernel of fuzzy colors, the sets of pixels painted in every fuzzy color are crisp, and hence a crisp cardinality is obtained. In particular, the fuzzy number for image A is the same despite the method employed for calculating it. In the gradual approach, crisp results are represented by the fact that only the level α = 1 is necessary, since all the levels are assigned the same cardinality (in this case, 1/8). In the case of the quantifier, any quantifier having 1/8 in its core will give a result of 1 in the evaluation of the quantified sentence. We assume here that the user has predefined a collection of quantifiers of

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Fig. 2. Two synthetic images. (For interpretation of the colors in this figure, the reader is referred to the web version of this article.) Table 2 1 , . . . , C 8 ) in the Color Histograms obtained for images Cardinalities (same for all fuzzy colors C A and B in Fig. 2 by using different approaches to cardinality. Image A

Image B

Quantifier Gradual

Around 12.5% 1 → 1/8

Fuzzy number Scalar

1/(1/8) 1/8

Around 0% or Around 25% 1 → 0 0.5 →  1/4 1/0 + 0<x1/4 0.5/x 1/8

the form Around x%, with x ∈ [0, 1] corresponding to some kind of fuzzy partition of the unit interval. Under this assumption, Around 12.5% is a triangular fuzzy number/quantifier with kernel 1/8 with appropriate specificity for the user. Of course, in this particular case, it is clear that a quantifier Exactly 12.5% would have been a better choice. In the case of image B, the first thing to highlight is the fact that the scalar cardinality yields the same result than that of image A. However, the perception of the colors that appear in the image and the corresponding frequencies is very different in both images, the only possible frequencies being intuitively 1/8 in image A, and 0 and 1/4 in image B. This example illustrates that the sigma-count is not an accurate measure in general. Notice also that using a threshold below 1/2 does not solve the problem. The fuzzy cardinality provided is the one that would be provided by most existing methods, and represents the notion of approximately Between 0 and 1/4. Though a suitable information for a human user, it is not an accurate representation, since there is no chance that the cardinality is other than 0 or 1/4. However, notice that it yields a different result than that for image A, and hence it is more accurate than the sigma-count. As a final remark, the fuzzy cardinality defined in [15] and employed in [36] gives a fuzzy set 0.5/0 + 0.5/(1/4) which is not a fuzzy number since it is neither normalized nor convex. This is more accurate than a fuzzy number and, though good enough to be easily understandable by a human, this is not always the case, as the fuzzy set can have a large support. Hence, it is a kind of compromise between accuracy and understandability. The gradual approach gives also a different result for both images, in this case representing accurately the only possible cardinalities in both images, associated to different levels. In addition, this representation is very easy to use for further calculations, since we have a crisp number in each level, and any kind of calculation to be employed is performed independently in each level following Definition 2.1. The results obtained in each level can later be summarized using techniques similar to those employed here to provide a quantifier from a gradual number. Finally, the evaluation of quantified sentences would provide for image B a fulfilment degree of 0.5 for both triangular quantifiers Around 0% and Around 25%. This is in consonance with our intuition about the possible cardinalities in image B. Here, two different options are to choose one of them on the basis of other application or user-specific information (for example, whether we prefer to be conservative or not with respect to the amount of colors), or to give both of them. Notice that this case is rather infrequent, and it is motivated by the fact that very specific, synthetic images with particular memberships have been employed. 4.2.4. A real example Let us consider the real image in Fig. 3. We have measured cardinality by means of gradual numbers, using 100 equidistributed values between 1 and 0.01. We are not showing the results of the gradual histogram here because of

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Fig. 3. Real image.

lack of space, being also little informative in a real case to the user. In order to obtain a linguistic histogram, we have considered a collection of ten triangular relative quantifiers defining a fuzzy partition in Ruspini’s sense, with kernels being the percentages 0, 1, 2, 3, 4, 12, 25, 50, 75, and 100. Let us denote by QPx the quantifier with kernel x. Tables 3  for and 4 show the values obtained by evaluation of the quantified sentences QPx of the pixels are painted in color C  in the fuzzy color space associated to the ISCC–NBS Basic set of colors explained in Section 4.1. every fuzzy color C We have employed the new method introduced in Section 3.3 for evaluating the sentence, obtaining the compatibility between the gradual number measuring the cardinality, and the quantifier. The last column in Table 4 contains the values of the linguistic histogram. Let us remark that the addition of the result of the evaluation of quantified sentences is not necessarily 1, since these are not frequencies but values of compatibility between cardinality and quantifiers. Similarly, the addition of the quantities indicated in the linguistic histogram is not expected to be 100% in general, since these are fuzzy sets around the value. Specificity also plays a role here, for example, the collection of quantifiers employed here “jump” from 4% to 12%, hence the latter quantifier is much less specific than those between 0–4%. In practice, we are just working on the basis of the quantifiers the user is interested in, his/her vocabulary, and the results may vary depending on the number and definition of quantifiers, and the fuzzy color space employed. However, we think that the results are compatible with what we can see in the image. Finally, choosing more than one quantifier when the difference in the accomplishment degree is very low can be an interesting alternative in some cases, particularly when there are several quantifiers with a compatibility degree close to the maximum. 4.3. Color dominance Color dominance is an important concept in image processing. The dominant color descriptor is one of the most important descriptors in MPEG-7. The notion of dominance is basically related to the frequency of the color in the image, though other aspects may be also taken into account. A dominant color descriptor must provide an effective, compact, and intuitive representation of the most representative (frequent) colors present in an image. Many approaches to dominant color extraction have been proposed in the literature [50,30]. Most of them perform the extraction process based on histogram analysis [42,53] or clustering techniques [29] in color domain. Nevertheless, these approaches consider a crisp notion of dominance, when in fact for human’s perception there are degrees of dominance, that is, colors can be clearly dominant, clearly not dominant, or can be dominant to a certain degree. In addition, most of the times they do not consider subsets of crisp colors, as represented in computers, that fully match human perception as expressed by linguistic color terms. We have proposed to define the fuzzy concept dominant via quantifiers in previous works [9,8,12]. A fuzzy nondecreasing quantifier is a natural way to represent the semantics of the concept on the basis of the amount of pixels having a certain color.

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Table 3 Evaluation of quantified sentences for the image in Fig. 3. Fuzzy color  C

Quantifiers QP0

QP1

QP2

QP3

QP4

QP12

Pink Red Orange Brown Yellow Olive Yellow-green Green Blue Purple White Gray Black

0 0 0 0 0 0 0.39 0 0 0 0 0 0

0.01 0 0 0 0 0.01 0.19 0.01 0.05 0.01 0.1 0.05 0

0.02 0 0 0 0 0.06 0.17 0.03 0.05 0.01 0.21 0.06 0

0.04 0 0 0.02 0 0.06 0.15 0.03 0.04 0.02 0.22 0.06 0.02

0.3 0 0 0.28 0.07 0.37 0.18 0.23 0.28 0.17 0.31 0.24 0.26

0.47 0.54 0.33 0.52 0.59 0.38 0.06 0.49 0.41 0.43 0.09 0.38 0.51

Table 4 Evaluation of quantified sentences and linguistic color histogram for the image in Fig. 3. Fuzzy color  C

Quantifiers QP25

QP50

QP75

QP100

Linguistic histogram  hL (C)

Pink Red Orange Brown Yellow Olive Yellow-green Green Blue Purple White Gray Black

0.18 0.44 0.4 0.18 0.1 0.11 0 0.2 0.11 0.32 0 0.19 0.08

0 0.01 0.09 0 0 0 0 0 0 0.03 0 0.01 0

0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0

approximately 12% approximately 12% approximately 25% approximately 12% approximately 12% approximately 12% approximately 0% approximately 12% approximately 12% approximately 12% approximately 4% approximately 12% approximately 12%

Table 5 Definition of quantifiers for representing “approximately x or more” for different values of x. Parameter x

Trapezoidal quantifier

0.02 0.04 0.06 0.1 0.2 0.4 0.5

(0.01, 0.02, 1, 1) (0.03, 0.04, 1, 1) (0.05, 0.06, 1, 1) (0.06, 0.1, 1, 1) (0.1, 0.2, 1, 1) (0.2, 0.4, 1, 1) (0.4, 0.5, 1, 1)

We show here an example for the image in Fig. 3, using the same cardinality measurement as in the case of histograms. In this case, we have employed a collection of non-decreasing quantifiers, each one corresponding to a different notion of dominance as “approximately x or more”. The definition of each quantifier as a trapezoidal fuzzy number has been made as shown in Table 5. We have used again the evaluation method introduced in Section 3.3. The results are shown in Table 6, where each column represents the degrees of dominance we may obtain for each of the possible quantifiers that we have considered. The number on top of each column is the parameter x and the corresponding quantifier is that shown

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Table 6 Color dominance as defined by different quantifiers for the image in Fig. 3. Fuzzy color  C

0.02

Quantifiers (parameter x) 0.04

0.06

0.1

0.2

0.4

0.5

Pink Red Orange Brown Yellow Olive Yellow-green Green Blue Purple White Gray Black

0.89 1 1 1 1 0.8 0.39 0.85 0.69 0.91 0.65 0.74 1

0.72 1 1 0.84 0.93 0.61 0.18 0.72 0.58 0.8 0.31 0.57 0.85

0.59 0.92 0.86 0.69 0.82 0.48 0.02 0.61 0.49 0.7 0.05 0.47 0.71

0.46 0.78 0.64 0.52 0.61 0.36 0 0.49 0.39 0.55 0 0.39 0.52

0.25 0.54 0.46 0.26 0.21 0.17 0 0.27 0.18 0.39 0 0.24 0.19

0.01 0.15 0.2 0 0 0 0 0 0 0.13 0 0.07 0

0 0 0 0 0 0 0 0 0 0 0 0 0

in Table 5. Of course, in a certain application, a single quantifier is employed for defining the notion of dominance. In this case, we have chosen an image in which almost every possible color is rather frequent, and hence the values obtained are rather high. As expected, as the definition of dominance is more restrictive (larger x), degrees of dominance are lower, and are 0 for every color when we consider “approximately 50% or more” as the condition for a color to be dominant. 5. Conclusions and future work Quantified sentences play a very important role in image processing in order to define semantic concepts. Quantification relies on cardinality, and hence it is necessary to have good cardinality measures. We have discussed scalar, fuzzy, and gradual numbers as measurements of the cardinality of fuzzy sets. We have determined two different tasks that require a separate treatment: providing an accurate measurement of cardinality, and providing a linguistic or at least clearly understandable information about cardinality to a human user. We have concluded that scalar measures are not well suited neither as measures of cardinality nor as informative summaries for human users, since they are neither accurate nor representative in general. Fuzzy numbers are well suited, and the best choice in our opinion, for providing linguistic information to the user; however, they are not good as measurements since the possible cardinalities of a fuzzy set are those of its α-cuts, and this leads to non-convexity of cardinality when represented as a fuzzy subset of the non-negative integers. Finally, gradual numbers, introduced by Dubois and Prade [21], are the best choice in our opinion for measuring and arithmetics. However, they are not well suited for providing linguistic information to the user. On this basis, our proposal is to employ gradual numbers for measuring cardinality and for performing further calculations on its basis, and to employ fuzzy numbers in a user-specific vocabulary of linguistic labels for providing linguistic information. For that purpose, we propose to calculate the compatibility between a gradual number and a fuzzy number, corresponding to the evaluation of a quantified sentence in which the fuzzy number is the quantifier, and then to take the fuzzy number with higher compatibility. We have discussed about different methods for evaluating quantified sentences, and we have proposed a new one. We have illustrated the possibilities of the new proposal in two applications in image processing, among the many possibilities in this field. First, the calculation of color histograms. We propose to use gradual histograms when the result is to be employed as the basis for further calculation, whilst we consider linguistic histograms to be more appropriate when the information is to be given to the user. Second, the definition of the semantics of the concept of dominance for color. As future work we plan to use histograms and dominance as defined here in developing applications for linguistic description of images and image information retrieval. We will also employ our cardinality and quantification schemes for defining other semantic concepts, like those shown in the introduction, and many others. Finally, we will study

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